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1、1
2、ClassicalSymmetriesTheconceptofsymmetrywillplayacrucialroleinnearlyallaspectsofourdiscussionofweakinteractions.Atthelevelofthedynamics,thefundamentalinteractions(oratleastthatsubsetofthefundamentalinteractionsthatweunderstand)areassociatedwithgauge
3、symmetries".Butmorethanthat,theunderlyingmathematicallanguageofrelativisticquantummechanics
4、quan-tumeldtheory
5、ismucheasiertounderstandifyoumakeuseofallthesymmetryinformationthatisavailable.Inthiscourse,wewilmakeextensiveuseofsymmetryasamathematicalto
6、oltohelpusunderstandthephysics.Inparticular,wemakeuseofthelanguageofrepresentationsofLiealgebras.1.1Noether'sTheorem{ClassicalAttheclassicallevel,symmetriesofanactionwhichisanintegralofalocalLagrangiandensityareassociatedwithconservedcurrents.Consider
7、asetofelds,(x)wherej=1toN,andanactionjZ4S[]=dxL((x)@(x))(1.1.1)whereListhelocalLagrangiandensity.Theindex,j,iswhatparticlephysicistscalla
avor"index.Dierentvaluesofjlabeldierenttypes,or
avors",oftheeld.Thinkoftheeld,,withoutanyexpliciti
8、ndex,asacolumnvectorin
avorspace.Assume,forsimplicity,thattheLagrangiandependsonlyontheelds,,andtheirrstderivatives,@.TheequationsofmotionareLL@=:(1.1.2)(@)Notethat(1.1.2)isavectorequationin
avorspace.Eachsideisarowvector,carryingthe
avor
9、index,j.Asymmetryoftheactionissomeinnitesimalchangeintheelds,,suchthatS[+]=S[](1.1.3)orL(+@+@)=L(@)+@V(@)(1.1.4)whereVissomevectorfunctionoftheorderoftheinnitesimal,.Weassumeherethatwecan4throwawaysurfacetermsinthe
10、dxintegralsothattheVtermsmakesnocontributiontotheaction.ButLLL(+@+@);L(@)=+@(1.1.5)(@)1WeakInteractions
11、HowardGeorgi
12、draft-February10,1998
13、2because@=@.Notethat(1.1.5)isasingleequationwithnojindex.Thetermsontherighthands
14、ideinvolveamatrixmultiplicationin
avorspaceofarowvectorontheleftwithacolumnvectorontheright.From(1.1.2),(1.1.4)and(1.1.5),wehave@N=0(1.1.6)whereLN=;V:(1.1.7)(@)Often,wewilbeinterestedinsymmetriesthataresymmetriesoftheLagrangian,notj